Cheaper alternative to visiting Poland with children?

My husband is a Polish ex-pat, now a US citizen. We have 3 small children (7 yr old twins and a 2-yr old). He still has family living in Poland, in Gdansk. We've been discussing visiting for a bit over the summer but have run into a major stumbling block in that tickets are very, very expensive - at least $6000 round trip for the five of us.

Is there a cheaper alternative than flying into Warsaw? Both he and I are quite comfortable in Europe, so we would have no problem flying into Paris, Frankfurt, etc. and then taking the train. But I don't know if that would be expensive and/or inconvenient enough that it would just be better to fly straight to Warsaw.

When we first got married, LOT had a special for families - basically parents pay, children go free. Too bad they don't have anything similar...

Maybe flying to UK first? There are many "low cost" flights to Poland.

Flying to the UK won't make a lot of difference (maybe saving of 1K total) - but with children it is much more difficult to plan. I once had a good deal from SAS airlines - Chicago through Kopenhagen and Warsaw.

Going via Germany seems popular. Sorry, can't add more.